1,212 research outputs found
Quantum extended crystal PDE's
Our recent results on {\em extended crystal PDE's} are generalized to PDE's
in the category of quantum supermanifolds. Then obstructions
to the existence of global quantum smooth solutions for such equations are
obtained, by using algebraic topologic techniques. Applications are considered
in details to the quantum super Yang-Mills equations. Furthermore, our
geometric theory of stability of PDE's and their solutions, is also generalized
to quantum extended crystal PDE's. In this way we are able to identify quantum
equations where their global solutions are stable at finite times. These
results, are also extended to quantum singular (super)PDE's, introducing ({\em
quantum extended crystal singular (super) PDE's}).Comment: 43 pages, 1 figur
Towards a definition of quantum integrability
We briefly review the most relevant aspects of complete integrability for
classical systems and identify those aspects which should be present in a
definition of quantum integrability.
We show that a naive extension of classical concepts to the quantum framework
would not work because all infinite dimensional Hilbert spaces are unitarily
isomorphic and, as a consequence, it would not be easy to define degrees of
freedom. We argue that a geometrical formulation of quantum mechanics might
provide a way out.Comment: 37 pages, AmsLatex, 1 figur
Newton-Hooke type symmetry of anisotropic oscillators
The rotation-less Newton--Hooke - type symmetry found recently in the Hill
problem and instrumental for explaining the center-of-mass decomposition is
generalized to an arbitrary anisotropic oscillator in the plane. Conversely,
the latter system is shown, by the orbit method, to be the most general one
with such a symmetry. Full Newton-Hooke symmetry is recovered in the isotropic
case. Star escape from a Galaxy is studied as application.Comment: Updated version with more figures added. 34 pages, 7 figures.
Dedicated to the memory of J.-M. Souriau, deceased on March 15 2012, at the
age of 9
Superintegrable Deformations of the Smorodinsky-Winternitz Hamiltonian
A constructive procedure to obtain superintegrable deformations of the
classical Smorodinsky-Winternitz Hamiltonian by using quantum deformations of
its underlying Poisson sl(2) coalgebra symmetry is introduced. Through this
example, the general connection between coalgebra symmetry and quasi-maximal
superintegrability is analysed. The notion of comodule algebra symmetry is also
shown to be applicable in order to construct new integrable deformations of
certain Smorodinsky-Winternitz systems.Comment: 17 pages. Published in "Superintegrability in Classical and Quantum
Systems", edited by P.Tempesta, P.Winternitz, J.Harnad, W.Miller Jr.,
G.Pogosyan and M.A.Rodriguez, CRM Proceedings & Lecture Notes, vol.37,
American Mathematical Society, 200
A systematic construction of completely integrable Hamiltonians from coalgebras
A universal algorithm to construct N-particle (classical and quantum)
completely integrable Hamiltonian systems from representations of coalgebras
with Casimir element is presented. In particular, this construction shows that
quantum deformations can be interpreted as generating structures for integrable
deformations of Hamiltonian systems with coalgebra symmetry. In order to
illustrate this general method, the algebra and the oscillator
algebra are used to derive new classical integrable systems including a
generalization of Gaudin-Calogero systems and oscillator chains. Quantum
deformations are then used to obtain some explicit integrable deformations of
the previous long-range interacting systems and a (non-coboundary) deformation
of the Poincar\'e algebra is shown to provide a new
Ruijsenaars-Schneider-like Hamiltonian.Comment: 26 pages, LaTe
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